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August  2022, 42(8): 3787-3807. doi: 10.3934/dcds.2022032

Maximal regularity for time-stepping schemes arising from convolution quadrature of non-local in time equations

1. 

Universidad de Santiago de Chile, Departamento de Matemática y Ciencia de la Computación, Las Sophoras 173, Estación Central, Santiago, Chile

2. 

Universitat Politècnica de València, Instituto Universitario de Matemática Pura y Aplicada (IUMPA), Camino de Vera s/n, 46022 València, Spain

* Corresponding author: Carlos Lizama

Received  September 2021 Revised  February 2022 Published  August 2022 Early access  March 2022

We study discrete time maximal regularity in Lebesgue spaces of sequences for time-stepping schemes arising from Lubich's convolution quadrature method. We show minimal properties on the quadrature weights that determines a wide class of implicit schemes. For an appropriate choice of the weights, we are able to identify the $ \theta $-method as well as the backward differentiation formulas and the $ L1 $-scheme. Fractional versions of these schemes, some of them completely new, are also shown, as well as their representation by means of the Grünwald–Letnikov fractional order derivative. Our results extend and improve some recent results on the subject and provide new insights on the basic nature of the weights that ensure maximal regularity.

Citation: Carlos Lizama, Marina Murillo-Arcila. Maximal regularity for time-stepping schemes arising from convolution quadrature of non-local in time equations. Discrete and Continuous Dynamical Systems, 2022, 42 (8) : 3787-3807. doi: 10.3934/dcds.2022032
References:
[1]

R. P. Agarwal, C. Cuevas and C. Lizama, Regularity of Difference Equations on Banach Spaces, Springer-Verlag, Cham, 2014. doi: 10.1007/978-3-319-06447-5.

[2]

G. Akrivis and E. Katsoprinakis, Maximum angles of $A(\theta)$-stability of backward difference formulae, BIT Numer. Math., 60 (2020), 93-99.  doi: 10.1007/s10543-019-00768-1.

[3]

G. AkrivisB. Li and C. Lubich, Combining maximal regularity and energy estimates for time discretizations of quasilinear parabolic equations, Math. Comp., 86 (2017), 1527-1552.  doi: 10.1090/mcom/3228.

[4]

H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory, Monographs in Mathematics, 89. Birkhäuser-Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9221-6.

[5]

F. M. Atici and P. W. Eloe, Initial value problems in discrete fractional calculus, Proc. Amer. Math. Soc., 137 (2009), 981-989.  doi: 10.1090/S0002-9939-08-09626-3.

[6]

S. Blunck, Analyticity and discrete maximal regularity on Lp-spaces, J. Funct. Anal., 183 (2001), 211-230.  doi: 10.1006/jfan.2001.3740.

[7]

S. Blunck, Maximal regularity of discrete and continuous time evolution equations, Studia Math., 146 (2001), 157-176.  doi: 10.4064/sm146-2-3.

[8]

M. Caputo, Diffusion of fluids in porous media with memory, Geothermics, 28 (1999), 113-130.  doi: 10.1016/S0375-6505(98)00047-9.

[9]

E. CuestaC. Lubich and C. Palencia, Convolution quadrature time discretization of fractional diffusion-wave equations, Math. Comp., 75 (2006), 673-696.  doi: 10.1090/S0025-5718-06-01788-1.

[10]

R. Denk, M. Hieber and J. Prüss, $\mathcal{R}$-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166 (2003). doi: 10.1090/memo/0788.

[11]

P. Flajolet, Singularity analysis and asymptotics of Bernoulli sums, Theoret. Comput. Sci., 215 (1999), 371-381.  doi: 10.1016/S0304-3975(98)00220-5.

[12]

C. S. Goodrich and C. Lizama, A transference principle for nonlocal operators using a convolutional approach: Fractional monotonicity and convexity, Israel J. Math., 236 (2020), 533-589.  doi: 10.1007/s11856-020-1991-2.

[13]

B. JinR. Lazarov and Z. Zhou, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA Journal of Numerical Analysis, 36 (2016), 197-221.  doi: 10.1093/imanum/dru063.

[14]

B. JinB. Li and Z. Zhou, Discrete maximal regularity of time-stepping schemes for fractional evolution equations, Numer. Math., 138 (2018), 101-131.  doi: 10.1007/s00211-017-0904-8.

[15]

T. Kemmochi, Discrete maximal regularity for abstract Cauchy problems, Studia Math., 234 (2016), 241-263.  doi: 10.4064/sm8495-7-2016.

[16]

P. Klouček and F. S. Rys, Stability of the fractional step $\theta$-scheme for the nonstationary Navier–Stokes equations, SIAM J. Numer. Anal., 31 (1994), 1312-1335.  doi: 10.1137/0731068.

[17]

A. N. Kochubei, General fractional calculus, evolution equations, and renewal processes, Integr. Equ. Oper. Theory, 71 (2011), 583-600.  doi: 10.1007/s00020-011-1918-8.

[18]

B. KovácsB. Li and C. Lubich, A-stable time discretizations preserve maximal parabolic regularity, SIAM J. Numer. Anal., 54 (2016), 3600-3624.  doi: 10.1137/15M1040918.

[19]

D. LeykekhmanB. Vexler and D. Walter, Numerical analysis of sparse initial data identification for parabolic problems, ESAIM Math. Model. Numer. Anal., 54 (2020), 1139-1180.  doi: 10.1051/m2an/2019083.

[20]

B. Li and W. Sun, Regularity of the diffusion-dispersion tensor and error analysis of FEMs for a porous media flow, SIAM J. Numer. Anal., 53 (2015), 1418-1437.  doi: 10.1137/140958803.

[21]

C. Lizama, The Poisson distribution, abstract fractional difference equations, and stability, Proc. Amer. Math. Soc., 145 (2017), 3809-3827.  doi: 10.1090/proc/12895.

[22]

C. Lizama, $\ell_p$-maximal regularity for fractional difference equations on $UMD$ spaces, Math. Nach., 288 (2015), 2079-2092.  doi: 10.1002/mana.201400326.

[23]

C. Lizama and M. Murillo-Arcila, Discrete maximal regularity for Volterra equations and nonlocal time-stepping schemes, Discrete Contin. Dyn. Syst. Series A, 40 (2020), 509-528.  doi: 10.3934/dcds.2020020.

[24]

C. Lubich, Convolution quadrature and discretized operational calculus. I., Numer. Math., 52 (1988), 129-145.  doi: 10.1007/BF01398686.

[25]

M. D. OrtigueiraF. J. V. Coito and J. J. Trujillo, Discrete-time differential systems, Signal Processing, 107 (2015), 198-217. 

[26]

P. Portal, Discrete time analytic semigroups and the geometry of Banach spaces, Semigroup Forum, 67 (2003), 125-144.  doi: 10.1007/s00233-002-0009-1.

[27]

J. Prüss, Evolutionary Integral Equations and Applications, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 1993. doi: 10.1007/978-3-0348-8570-6.

[28]

S. G. Samko and R. P. Cardoso, Integral equations of the first kind of Sonine type, Int. J. Math. and Math. Sci., 57 (2003), 3609-3632.  doi: 10.1155/S0161171203211455.

[29]

E. Süli, Numerical Solution of Ordinary Differential Equations, Mathematical Institute, Oxford University, 2014.

[30]

Y. Sun and P. C. B. Phillips, Understanding the Fisher equation, J. Appl. Econ., 19 (2004), 869-886.  doi: 10.1002/jae.760.

[31]

Z.-Z. Sun and X. Wu, A fully discrete scheme for a diffusion-wave system, Appl. Numer. Math., 56 (2006), 193-209.  doi: 10.1016/j.apnum.2005.03.003.

[32]

V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers. Volume I. Background and Theory, Nonlinear Physical Science, Higher Education Press, Beijing; Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-33911-0.

[33]

V. Vergara and R. Zacher, Optimal decay estimates for time-fractional and other non-local subdiffusion equations via energy methods, SIAM J. Math. Anal., 47 (2013), 210-239.  doi: 10.1137/130941900.

[34]

L. Weis, Operator-valued Fourier multiplier theorems and maximal $L_p$-regularity, Math. Ann., 319 (2001), 735-758.  doi: 10.1007/PL00004457.

[35]

D. Wood, The Computation of Polylogarithms, Technical Report 15-92, University of Kent, Computing Laboratory, University of Kent, Canterbury, UK, June 1992.

show all references

References:
[1]

R. P. Agarwal, C. Cuevas and C. Lizama, Regularity of Difference Equations on Banach Spaces, Springer-Verlag, Cham, 2014. doi: 10.1007/978-3-319-06447-5.

[2]

G. Akrivis and E. Katsoprinakis, Maximum angles of $A(\theta)$-stability of backward difference formulae, BIT Numer. Math., 60 (2020), 93-99.  doi: 10.1007/s10543-019-00768-1.

[3]

G. AkrivisB. Li and C. Lubich, Combining maximal regularity and energy estimates for time discretizations of quasilinear parabolic equations, Math. Comp., 86 (2017), 1527-1552.  doi: 10.1090/mcom/3228.

[4]

H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory, Monographs in Mathematics, 89. Birkhäuser-Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9221-6.

[5]

F. M. Atici and P. W. Eloe, Initial value problems in discrete fractional calculus, Proc. Amer. Math. Soc., 137 (2009), 981-989.  doi: 10.1090/S0002-9939-08-09626-3.

[6]

S. Blunck, Analyticity and discrete maximal regularity on Lp-spaces, J. Funct. Anal., 183 (2001), 211-230.  doi: 10.1006/jfan.2001.3740.

[7]

S. Blunck, Maximal regularity of discrete and continuous time evolution equations, Studia Math., 146 (2001), 157-176.  doi: 10.4064/sm146-2-3.

[8]

M. Caputo, Diffusion of fluids in porous media with memory, Geothermics, 28 (1999), 113-130.  doi: 10.1016/S0375-6505(98)00047-9.

[9]

E. CuestaC. Lubich and C. Palencia, Convolution quadrature time discretization of fractional diffusion-wave equations, Math. Comp., 75 (2006), 673-696.  doi: 10.1090/S0025-5718-06-01788-1.

[10]

R. Denk, M. Hieber and J. Prüss, $\mathcal{R}$-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166 (2003). doi: 10.1090/memo/0788.

[11]

P. Flajolet, Singularity analysis and asymptotics of Bernoulli sums, Theoret. Comput. Sci., 215 (1999), 371-381.  doi: 10.1016/S0304-3975(98)00220-5.

[12]

C. S. Goodrich and C. Lizama, A transference principle for nonlocal operators using a convolutional approach: Fractional monotonicity and convexity, Israel J. Math., 236 (2020), 533-589.  doi: 10.1007/s11856-020-1991-2.

[13]

B. JinR. Lazarov and Z. Zhou, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA Journal of Numerical Analysis, 36 (2016), 197-221.  doi: 10.1093/imanum/dru063.

[14]

B. JinB. Li and Z. Zhou, Discrete maximal regularity of time-stepping schemes for fractional evolution equations, Numer. Math., 138 (2018), 101-131.  doi: 10.1007/s00211-017-0904-8.

[15]

T. Kemmochi, Discrete maximal regularity for abstract Cauchy problems, Studia Math., 234 (2016), 241-263.  doi: 10.4064/sm8495-7-2016.

[16]

P. Klouček and F. S. Rys, Stability of the fractional step $\theta$-scheme for the nonstationary Navier–Stokes equations, SIAM J. Numer. Anal., 31 (1994), 1312-1335.  doi: 10.1137/0731068.

[17]

A. N. Kochubei, General fractional calculus, evolution equations, and renewal processes, Integr. Equ. Oper. Theory, 71 (2011), 583-600.  doi: 10.1007/s00020-011-1918-8.

[18]

B. KovácsB. Li and C. Lubich, A-stable time discretizations preserve maximal parabolic regularity, SIAM J. Numer. Anal., 54 (2016), 3600-3624.  doi: 10.1137/15M1040918.

[19]

D. LeykekhmanB. Vexler and D. Walter, Numerical analysis of sparse initial data identification for parabolic problems, ESAIM Math. Model. Numer. Anal., 54 (2020), 1139-1180.  doi: 10.1051/m2an/2019083.

[20]

B. Li and W. Sun, Regularity of the diffusion-dispersion tensor and error analysis of FEMs for a porous media flow, SIAM J. Numer. Anal., 53 (2015), 1418-1437.  doi: 10.1137/140958803.

[21]

C. Lizama, The Poisson distribution, abstract fractional difference equations, and stability, Proc. Amer. Math. Soc., 145 (2017), 3809-3827.  doi: 10.1090/proc/12895.

[22]

C. Lizama, $\ell_p$-maximal regularity for fractional difference equations on $UMD$ spaces, Math. Nach., 288 (2015), 2079-2092.  doi: 10.1002/mana.201400326.

[23]

C. Lizama and M. Murillo-Arcila, Discrete maximal regularity for Volterra equations and nonlocal time-stepping schemes, Discrete Contin. Dyn. Syst. Series A, 40 (2020), 509-528.  doi: 10.3934/dcds.2020020.

[24]

C. Lubich, Convolution quadrature and discretized operational calculus. I., Numer. Math., 52 (1988), 129-145.  doi: 10.1007/BF01398686.

[25]

M. D. OrtigueiraF. J. V. Coito and J. J. Trujillo, Discrete-time differential systems, Signal Processing, 107 (2015), 198-217. 

[26]

P. Portal, Discrete time analytic semigroups and the geometry of Banach spaces, Semigroup Forum, 67 (2003), 125-144.  doi: 10.1007/s00233-002-0009-1.

[27]

J. Prüss, Evolutionary Integral Equations and Applications, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 1993. doi: 10.1007/978-3-0348-8570-6.

[28]

S. G. Samko and R. P. Cardoso, Integral equations of the first kind of Sonine type, Int. J. Math. and Math. Sci., 57 (2003), 3609-3632.  doi: 10.1155/S0161171203211455.

[29]

E. Süli, Numerical Solution of Ordinary Differential Equations, Mathematical Institute, Oxford University, 2014.

[30]

Y. Sun and P. C. B. Phillips, Understanding the Fisher equation, J. Appl. Econ., 19 (2004), 869-886.  doi: 10.1002/jae.760.

[31]

Z.-Z. Sun and X. Wu, A fully discrete scheme for a diffusion-wave system, Appl. Numer. Math., 56 (2006), 193-209.  doi: 10.1016/j.apnum.2005.03.003.

[32]

V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers. Volume I. Background and Theory, Nonlinear Physical Science, Higher Education Press, Beijing; Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-33911-0.

[33]

V. Vergara and R. Zacher, Optimal decay estimates for time-fractional and other non-local subdiffusion equations via energy methods, SIAM J. Math. Anal., 47 (2013), 210-239.  doi: 10.1137/130941900.

[34]

L. Weis, Operator-valued Fourier multiplier theorems and maximal $L_p$-regularity, Math. Ann., 319 (2001), 735-758.  doi: 10.1007/PL00004457.

[35]

D. Wood, The Computation of Polylogarithms, Technical Report 15-92, University of Kent, Computing Laboratory, University of Kent, Canterbury, UK, June 1992.

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